9/20/2018
University of Arkansas Statistics Seminar
Introduction
- Climate change is well understood globally.
- Climate change is less well understood locally.
- Need for spatailly explicit reconstructions of climate variables.
- Problem: data souruces are messy and noisy.
Introduction
Predicting the future by learning from the past
Introduction
Predicting the future by learning from the past
Introduction
Predicting the future by learning from the past
Introduction
Predicting the future by learning from the past
Learning about the past
Climate proxy data
- Many ecological and phyiscal processes respond to climate over different time scales.
- Tree rings, corals, forest landscapes, ice rings, lake levels, etc.
- These processes are called climate proxies.
- They are proxy measurements for unobserved climate.
- Noisy and messy.
- Respond to a wide variety of non-climatic signals.
Pollen Data
Pollen Data
Pollen Data
Fossil Pollen Data
Model Framework
Bayesian hierarchical model.
\(\begin{align*} [\mathbf{Z}, \boldsymbol{\theta}_D, \boldsymbol{\theta}_P | \mathbf{y}] & \propto [\mathbf{y} | \boldsymbol{\theta}_D, \mathbf{Z}] [\mathbf{Z} | \boldsymbol{\theta}_P] [\boldsymbol{\theta}_D] [\boldsymbol{\theta}_P] \end{align*}\)
Model Framework
Bayesian hierarchical model.
\(\begin{align*} \color{cyan}{[\mathbf{Z}, \boldsymbol{\theta}_D, \boldsymbol{\theta}_P | \mathbf{y}]} & \propto [\mathbf{y} | \boldsymbol{\theta}_D, \mathbf{Z}] [\mathbf{Z} | \boldsymbol{\theta}_P] [\boldsymbol{\theta}_D] [\boldsymbol{\theta}_P] \end{align*}\)
Posterior.
Model Framework
Bayesian hierarchical model.
\(\begin{align*} \color{cyan}{[\mathbf{Z}, \boldsymbol{\theta}_D, \boldsymbol{\theta}_P | \mathbf{y}]} & \propto \color{red}{[\mathbf{y} | \boldsymbol{\theta}_D, \mathbf{Z}]} [\mathbf{Z} | \boldsymbol{\theta}_P] [\boldsymbol{\theta}_D] [\boldsymbol{\theta}_P] \end{align*}\)
Posterior.
Data Model
Model Framework
Bayesian hierarchical model.
\(\begin{align*} \color{cyan}{[\mathbf{Z}, \boldsymbol{\theta}_D, \boldsymbol{\theta}_P | \mathbf{y}]} & \propto \color{red}{[\mathbf{y} | \boldsymbol{\theta}_D, \mathbf{Z}]} \color{blue}{[\mathbf{Z} | \boldsymbol{\theta}_P]} [\boldsymbol{\theta}_D] [\boldsymbol{\theta}_P] \end{align*}\)
Posterior.
Data Model.
Process Model.
Model Framework
Bayesian hierarchical model.
\(\begin{align*} \color{cyan}{[\mathbf{Z}, \boldsymbol{\theta}_D, \boldsymbol{\theta}_P | \mathbf{y}]} & \propto \color{red}{[\mathbf{y} | \boldsymbol{\theta}_D, \mathbf{Z}]} \color{blue}{[\mathbf{Z} | \boldsymbol{\theta}_P]} \color{orange}{[\boldsymbol{\theta}_D] [\boldsymbol{\theta}_P]} \end{align*}\)
Posterior.
Data Model.
Process Model.
Prior Model.
Data Model
\(\begin{align*} [\mathbf{Z}, \boldsymbol{\theta}_D, \boldsymbol{\theta}_P | \mathbf{y}] & \propto \color{red}{[\mathbf{y} | \boldsymbol{\theta}_D, \mathbf{Z}]} [\mathbf{Z} | \boldsymbol{\theta}_P] [\boldsymbol{\theta}_D] [\boldsymbol{\theta}_P] \end{align*}\)
Data model
- Describes how the data are collected and observed.
- Researchers take sediment samples from a lake.
- Take 1cm\(^3\) cubes along the length of the sediment core.
- In each cube, researcher counts the first \(N\) pollen grains and identifies to species.
- Raw data are counts of each species.
Data Model
For location \(\mathbf{s}\) and time \(t\),
\(\begin{align*} \mathbf{y} \left( \mathbf{s}_i, t \right) & = \left( y_{1} \left( \mathbf{s}_i, t \right), \ldots, y_{d} \left( \mathbf{s}_i, t \right) \right)' \end{align*}\)
is an observation of a \(d\)-dimensional compositional count.
- \(y_{j} \left( \mathbf{s}_i, t \right)\) is the count of species \(j\) in the sample at location \(\mathbf{s}_i\) and time \(t\).
- Compositional count data.
- Total count is not informative of the absolute composition.
- Informative of the relative proportions \(p_{j} \left( \mathbf{s}_i, t \right)\) only.
Data Model
- Compositional count vector \(\mathbf{y} \left( \mathbf{s}_i, t \right)\) a function of latent proportions \(\mathbf{p}\left( \mathbf{s}_i, t \right)\).
\(\begin{align*} \mathbf{y}\left( \mathbf{s}_i, t \right) | \mathbf{p}\left( \mathbf{s}_i, t \right) & \sim \operatorname{Multinomial} \left( N\left( \mathbf{s}_i, t \right), \mathbf{p}\left( \mathbf{s}_i, t \right) \right) \end{align*}\)
\(N\left( \mathbf{s}_i, t \right) = \sum_{j=1}^d y_{j}\left( \mathbf{s}_i, t \right)\) is the total count observed (fixed and known) for observation at location \(\mathbf{s}_i\) and time \(t\).
Compositional count vector \(\mathbf{y} \left( \mathbf{s}_i, t \right)\) a function of latent proportions \(\mathbf{p}\left( \mathbf{s}_i, t \right)\).
Overdispersion
- The pollen data are highly variable and overdispersed.
\(\begin{align*} \mathbf{p}\left( \mathbf{s}_i, t \right) | \boldsymbol{\alpha}\left( \mathbf{s}_i, t \right) & \sim \operatorname{Dirichlet} \left( \boldsymbol{\alpha}\left( \mathbf{s}_i, t \right) \right) \end{align*}\)
- Marginalize out \(\mathbf{p} \left( \mathbf{s}_i, t \right)\) to get Dirichlet-multinomial.
\(\begin{align*} \mathbf{y}\left( \mathbf{s}_i, t \right) | \boldsymbol{\alpha}\left( \mathbf{s}_i, t \right) & \sim \operatorname{Dirichlet-Multinomial} \left( N\left( \mathbf{s}_i, t \right), \boldsymbol{\alpha}\left( \mathbf{s}_i, t \right) \right) \end{align*}\)
Overdispersion
- We model the Dirichlet-multinomial count data using the log link function:
- \(\begin{align*} \operatorname{log} \left( \boldsymbol{\alpha} \left( \mathbf{s}_i, t \right) \right) & = \mathbf{z}\left( \mathbf{s}_i, t \right) \boldsymbol{\beta}. \end{align*}\)
- \(\mathbf{z}\left( \mathbf{s}_i, t \right)\)' is a \(q\)-dimensional vector of covariates.
- \(\boldsymbol{\beta}\) is a \(q \times d\) dimensional matrix of regression coefficients.
Calibration
- The \(\mathbf{z} \left( \mathbf{s}_i, t \right)\)s are observed only at \(t\) = 1.
- Calibration:
- Estimate \(\boldsymbol{\beta}\) using:
- \(\left( \mathbf{y} \left( \mathbf{s}_1, 1 \right), \ldots, \mathbf{y} \left( \mathbf{s}_n, 1 \right) \right)'\) and
- \(\left( \mathbf{z} \left( \mathbf{s}_1, 1 \right), \ldots, \mathbf{z} \left( \mathbf{s}_n, 1 \right) \right)'\).
- Estimate \(\boldsymbol{\beta}\) using:
- Reconstruction:
- Use estimated \(\boldsymbol{\beta}\)s and fossil pollen \(\mathbf{y} \left( \mathbf{s}, t \right)\) to predict unobserved \(\mathbf{z}\left( \mathbf{s}, t \right)\).
Calibration Model
Non-linear Calibration Model
- Vegetation response to climate is non-linear.
- \(\begin{align*} \operatorname{log} \left( \boldsymbol{\alpha} \left( \mathbf{s}_i, t \right) \right) & = \mathbf{B} \left( \mathbf{z}\left( \mathbf{s}_i, t \right) \right) \boldsymbol{\beta} \end{align*}\)
- \(\mathbf{B} \left( \mathbf{z}\left( \mathbf{s}_i, t \right) \right)\) is a basis expansion of the covariates \(\mathbf{z}\left( \mathbf{s}_i, t \right)\).
- Use B-splines or Gaussian Processes as a basis.
- Note that for \(t \neq 1\), the \(\mathbf{z} \left( \mathbf{s}_i, t \right)\)s are unobserved.
Non-linear Calibration Model
Process Model
\(\begin{align*} [\mathbf{Z}, \boldsymbol{\theta}_D, \boldsymbol{\theta}_P | \mathbf{y}] & \propto [\mathbf{y} | \boldsymbol{\theta}_D, \mathbf{Z}] \color{blue}{[\mathbf{Z} | \boldsymbol{\theta}_P]}[\boldsymbol{\theta}_D] [\boldsymbol{\theta}_P] \end{align*}\)
Process Model
Dynamic Model
- We are interested in estimating the latent process \(\mathbf{z} \left( \mathbf{s}, t \right)\).
- The model can accommodate:
- continuous vs. discrete space (geostatistical vs. CAR models).
- continuous vs. discrete time (stochastic process vs. AR models).
- For now, we focus on continuous space and discrete time.
Dynamic Model
- For \(\mathbf{z} \left(t \right) = \left( \mathbf{z} \left(\mathbf{s}_1, t \right)', \ldots, \mathbf{z} \left(\mathbf{s}_n, t \right)' \right)\), we assume:
\(\begin{align*} \mathbf{z} \left(t \right) - \mathbf{X} \left( t \right) \boldsymbol{\gamma} & = \mathbf{M}\left(t\right) \left( \mathbf{z} \left(t-1 \right) - \mathbf{X} \left( t \right) \boldsymbol{\gamma} \right) + \boldsymbol{\eta} \left(t \right) \end{align*}\)
- \(\mathbf{M}(t) = \rho \mathbf{I}_n\) is a propogator matrix.
- \(\mathbf{X} \left(t \right) \boldsymbol{\gamma}\) are the fixed effects from covariates like latitude, elevation, etc.
- \(\boldsymbol{\eta} \left( t \right) \sim \operatorname{N} \left( \mathbf{0}, \tau^2 \mathbf{R}\left( \boldsymbol{\phi} \right) \right)\).
- \(\tau^2\) is the spatial process variance.
- \(\mathbf{R} \left( \boldsymbol{\phi} \right)\) is a Mátern spatial covariance matrix with parameters \(\boldsymbol{\phi}\).
Elevation covariates
Time Uncertainty
- Each fossil pollen observation includes estimates of time uncertainty.
- The time of the observation is uncertain.
- Weight the likelihoods according to age-depth model.
- Posterior distribution of ages.
- For each observation fossil pollen observation an age-depth model gives a posterior distribution over dates.
- Define \(\omega \left(\mathbf{s}_i, t \right)\) as P(age \(\in (t-1, t)\)).
- \([\mathbf{y} \left( \mathbf{s}_i, t \right) | \boldsymbol{\alpha} \left( \mathbf{s}_i, t \right) ] = \prod_{t=1}^T [\mathbf{y} \left( \mathbf{s}_i, t \right) | \boldsymbol{\alpha} \left( \mathbf{s}, t \right)]^{\omega_\left(\mathbf{s}_i, t \right)}\).
Simulation Study
Simuated data
